The reality of existence of transfinite numbers

We can speak of the reality or the existence of integers in two ways, the finite and the infinite, but strictly speaking, any concepts or ideas that contemplate their reality apply equally to both. On the one hand, we may regard the integers as real insofar as they occupy a very specific place in our understanding on the basis of definitions, and are well-delineated from all our other mental concepts, and are in certain relationships with them and thus affect our thoughts in a certain way; allow me to call this kind of reality of our numbers their intra-subjective or fundamental reality.[Footnote13]

But reality can also be ascribed to the numbers insofar as they must be taken to be an expression or an image of the processes and relationships of the external world, as opposed to the mind, so that the various number-classes (I), (II), (III) and so on are representatives of cardinal-numbers that actually occur in the physical and the mental realm. I call this second type of reality the trans-subjective or ephemeral reality of the integers.

Given the completely realistic, but at the same time no less idealistic basis of my considerations, there is no doubt for me that these two types of reality always come together in the sense that a term that can be designated as existing in the first respect always has an ephemeral reality in a certain way, and even in infinitely many ways.[Footnote14]

This connection between the two realities has as its real reason the unity of the universe which we ourselves belong to. The purpose of my reference to this connection is to derive a conclusion that seems to me very important for mathematics, namely that it only takes into account the fundamental reality of its concepts in the formation of its ideas and is therefore not duty-bound to also examine them according to their ephemeral reality.

Because of this exceptional standpoint, which distinguishes it from all other sciences and which provides an explanation for the comparatively easy and unconstrained way of dealing with it, it particularly deserves the name of free mathematics, a designation which I would, if I had the choice, give preference to the term “pure” mathematics that has become common.

Mathematics is completely free in its development and is only bound to the self-evident consideration that its concepts are both free of contradictions and that they are in fixed relationships to proven concepts that have already been previously established.[Footnote15]

In particular, when new numbers are introduced, it is only necessary to give definitions of them which will afford them a sufficient definiteness and, under certain circumstances, such a relationship to the previously established numbers that they can be clearly distinguished from one another in given cases.

Once a number satisfies all these conditions, it can and must be considered as existing and real in mathematics. Regarding this one can now see, as indicated in § 4, the reason why one has to consider that the rational, irrational and complex numbers exist, in exactly the same way as we consider the finite positive integers to exist.

It is not necessary, I believe, to fear, as many do, any danger to science in these principles. On the one hand, the specified conditions under which the freedom of number formation can be given are such that they leave an extremely small scope for arbitrariness.

In addition every mathematical concept also carries the necessary corrective in itself; if it is sterile or inexpedient, it very soon shows it through its uselessness and it is then dropped because of lack of success. On the other hand, every superfluous constriction of the impulse for mathematical research seems to me to bring with it a much greater danger, and one that is all the greater as no justification can really be drawn for it from the nature of science; for the essence of mathematics lies precisely in its freedom.

Even if this characteristic of mathematics had not been recognized by me for the reasons mentioned, the entire development of science itself, as we perceive it in our century, would still have lead me to exactly the same views.

If Gauss, Cauchy, Abel, Jacobi, Dirichlet, Weierstrass, Hermite, and Riemann always had to subject their new ideas to the control of the philosophy of physics, we would certainly not have been able to enjoy the splendid structure of the modern theory of functions, which, although designed and established completely free of fixed purposes, has, as might be expected, already revealed its ephemeral meaning in applications to mechanics, astronomy and mathematical physics.

We would not have seen the great upswing in the theory of differential equations brought about by Fuchs, Poincare, and many others if these magnificent forces had been restrained and constricted by various influences. And if Kummer had not taken the momentous freedom of introducing so-called “ideal” numbers into number theory, we would not be in a position today to admire the important and exceptional algebraic and arithmetical works of Kronecker and Dedekind.

Therefore, although mathematics is entitled to move freely from all fetters of philosophy that relates to physics, on the other hand I cannot grant “applied” mathematics, such as analytical mechanics and mathematical physics, the same right.

These disciplines are, in my opinion, pertaining to the philosophy of physics both in their foundations and in their aims; if you seek to free yourself from this, as has recently been suggested by a famous physicist, you select a “description of nature” which must lack the fresh breath of free mathematical thought as well as the power to explain and explore natural phenomena.